Maths 212, Homework #1
First four problems:
due Thursday, Oct. 20
1. Let A = Q ∩ (0, π) be the set of all rational numbers between 0 and π. Find the least
upper bound of A and prove that your answer is correct.
2. Let A be a bounded set of real numbers and let B ⊂ A. Show that B is also bounded
and that sup B ≤ sup A. Establish a similar relation between inf A and inf B.
3. Define a sequence {sn } by setting s1 = 1 and then
sn+1 =
2sn + 2
sn + 2
for each n ≥ 1. Show that this sequence is convergent and find its limit.
4. Let {sn } be a convergent sequence and denote its limit by L. Given some L0 < L, show
that there exists an integer N such that sn > L0 for all n ≥ N .
5. (Squeeze law) Let {an }, {bn } and {cn } be three sequences with an ≤ bn ≤ cn for each
integer n. Assuming that an → L and cn → L, show that bn → L as well.
6. Let {sn } be a convergent sequence. Intuitively, sn+1 and sn should be approaching the
exact same limit. Verify this rigorously by showing that sn+1 − sn → 0.
7. Consider the function f : R → R defined by
¾
½
x
if x ∈ Q
.
f (x) =
0
if x ∈
/Q
Determine all points at which f is continuous.